Finding the Pattern in the Given Series

The given series is 6, 15, 35, 77, 143, ?
We need to find out the next number in the series.

Observations:
- The difference between the first two terms is 9 (15-6).
- The difference between the second and third terms is 20 (35-15).
- The difference between the third and fourth terms is 42 (77-35).
- The difference between the fourth and fifth terms is 66 (143-77).

Hence, we can conclude that the series is based on the concept of the square of the index number and adding the result to the initial term.

Calculating the Next Term in the Series

Using the given pattern, we can find out the next term in the series as follows:
- The initial term is 6.
- The square of the index number for the next term is 6^2 = 36.
- Adding the result to the initial term, we get the next term as 6 + 36 = 42.

Therefore, the next number in the given series is 42.

Verification of the Answer Options

Now, we can verify the answer options to find out which one is correct:
- Option A: 221 = 6 + 1^2 + 8^2 + 14^2 + 24^2 + 36^2 (Incorrect)
- Option B: 223 = 6 + 1^2 + 8^2 + 14^2 + 24^2 + 37^2 (Incorrect)
- Option C: 225 = 6 + 1^2 + 8^2 + 14^2 + 24^2 + 38^2 (Incorrect)
- Option D: 227 = 6 + 1^2 + 8^2 + 14^2 + 24^2 + 39^2 (Incorrect)

Therefore, the correct answer is option A) 221.